1,429 research outputs found
The difference between the domination number and the minus domination number of a cubic graph
AbstractThe closed neighborhood of a vertex subset S of a graph G = (V, E), denoted as N[S], is defined as the union of S and the set of all the vertices adjacent to some vertex of S. A dominating set of a graph G = (V, E) is defined as a set S of vertices such that N[S] = V. The domination number of a graph G, denoted as γ(G), is the minimum possible size of a dominating set of G. A minus dominating function on a graph G = (V, E) is a function g : V → {−1, 0, 1} such that g(N[v]) ≥ 1 for all vertices. The weight of a minus dominating function g is defined as g(V) =ΣvϵVg(v). The minus domination number of a graph G, denoted as γ−(G), is the minimum possible weight of a minus dominating function on G. It is well known that γ−(G) ≤ γ(G). This paper is focused on the difference between γ(G) and γ−(G) for cubic graphs. We first present a graph-theoretic description of γ−(G). Based on this, we give a necessary and sufficient condition for γ(G) −γ−(G) ≥ k. Further, we present an infinite family of cubic graphs of order 18k + 16 and with γ(G) −γ−(G) ≥
A note on domination and minus domination numbers in cubic graphs
Author name used in this publication: C. T. Ng2005-2006 > Academic research: refereed > Publication in refereed journalAccepted ManuscriptPublishe
Locating-dominating sets and identifying codes in graphs of girth at least 5
Locating-dominating sets and identifying codes are two closely related
notions in the area of separating systems. Roughly speaking, they consist in a
dominating set of a graph such that every vertex is uniquely identified by its
neighbourhood within the dominating set. In this paper, we study the size of a
smallest locating-dominating set or identifying code for graphs of girth at
least 5 and of given minimum degree. We use the technique of vertex-disjoint
paths to provide upper bounds on the minimum size of such sets, and construct
graphs who come close to meet these bounds.Comment: 20 pages, 9 figure
Location-domination in line graphs
A set of vertices of a graph is locating if every two distinct
vertices outside have distinct neighbors in ; that is, for distinct
vertices and outside , , where
denotes the open neighborhood of . If is also a dominating set (total
dominating set), it is called a locating-dominating set (respectively,
locating-total dominating set) of . A graph is twin-free if every two
distinct vertices of have distinct open and closed neighborhoods. It is
conjectured [D. Garijo, A. Gonzalez and A. Marquez, The difference between the
metric dimension and the determining number of a graph. Applied Mathematics and
Computation 249 (2014), 487--501] and [F. Foucaud and M. A. Henning.
Locating-total dominating sets in twin-free graphs: a conjecture. The
Electronic Journal of Combinatorics 23 (2016), P3.9] respectively, that any
twin-free graph without isolated vertices has a locating-dominating set of
size at most one-half its order and a locating-total dominating set of size at
most two-thirds its order. In this paper, we prove these two conjectures for
the class of line graphs. Both bounds are tight for this class, in the sense
that there are infinitely many connected line graphs for which equality holds
in the bounds.Comment: 23 pages, 2 figure
The domination parameters of cubic graphs
Let ir(G), γ(G), i(G), β0(G), Γ(G) and IR(G) be the irredundance number, the domination number, the independent domination number, the independence number, the upper domination number and the upper irredundance number of a graph G, respectively. In this paper we show that for any nonnegative integers k 1, k 2, k 3, k 4, k 5 there exists a cubic graph G satisfying the following conditions: γ(G) - ir(G) ≤ k 1, i(G) - γ(G) ≤ k 2, β0(G) - i(G) > k 3, Γ(G) - β0(G) - k 4, and IR(G) - Γ(G) - k 5. This result settles a problem posed in [9]. © Springer-Verlag 2005
The random geometry of equilibrium phases
This is a (long) survey about applications of percolation theory in
equilibrium statistical mechanics. The chapters are as follows:
1. Introduction
2. Equilibrium phases
3. Some models
4. Coupling and stochastic domination
5. Percolation
6. Random-cluster representations
7. Uniqueness and exponential mixing from non-percolation
8. Phase transition and percolation
9. Random interactions
10. Continuum modelsComment: 118 pages. Addresses: [email protected]
http://www.mathematik.uni-muenchen.de/~georgii.html [email protected]
http://www.math.chalmers.se/~olleh [email protected]
A Linear Kernel for Planar Total Dominating Set
A total dominating set of a graph is a subset such
that every vertex in is adjacent to some vertex in . Finding a total
dominating set of minimum size is NP-hard on planar graphs and W[2]-complete on
general graphs when parameterized by the solution size. By the meta-theorem of
Bodlaender et al. [J. ACM, 2016], there exists a linear kernel for Total
Dominating Set on graphs of bounded genus. Nevertheless, it is not clear how
such a kernel can be effectively constructed, and how to obtain explicit
reduction rules with reasonably small constants. Following the approach of
Alber et al. [J. ACM, 2004], we provide an explicit kernel for Total Dominating
Set on planar graphs with at most vertices, where is the size of the
solution. This result complements several known constructive linear kernels on
planar graphs for other domination problems such as Dominating Set, Edge
Dominating Set, Efficient Dominating Set, Connected Dominating Set, or Red-Blue
Dominating Set.Comment: 33 pages, 13 figure
Maker-Breaker total domination games on cubic graphs
We study Maker-Breaker total domination game played by two players, Dominator
and Staller on the connected cubic graphs. Staller (playing the role of Maker)
wins if she manages to claim an open neighbourhood of a vertex. Dominator wins
otherwise (i.e. if he can claim a total dominating set of a graph). For certain
graphs on vertices, we give the characterization on those which are
Dominator's win and those which are Staller's win
Progress towards the two-thirds conjecture on locating-total dominating sets
We study upper bounds on the size of optimum locating-total dominating sets
in graphs. A set of vertices of a graph is a locating-total dominating
set if every vertex of has a neighbor in , and if any two vertices
outside have distinct neighborhoods within . The smallest size of such a
set is denoted by . It has been conjectured that
holds for every twin-free graph of order
without isolated vertices. We prove that the conjecture holds for
cobipartite graphs, split graphs, block graphs, subcubic graphs and outerplanar
graphs
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